With a rubber mallet or brass punch, hammer the pinion gear. It will need to go in about ½” to ¾” to get past the pinion flange. The flange will fall off and a few more whacks should free up the pinion. Remove it from the housing and set aside.

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I assume that means to hit the side of the pinion gear towards the front of the car (that the nut goes on) and drive it back.

It looks like the flange is just held on by friction. I'm hammering, prying, soaking in oil, and doing everything else I can think of, but it's not coming off.

Do I have the right idea? Is there some trick? All I can think is to go buy a 6 foot breaker bar or get the sledge hammer out.

Take the pinion nut you just took off, and thread it back on the pinon till it is almost flush with the end of the threads. Almost like the nut is sticking out only a thread or so off the pinion gear. Then hit it with a sledge hammer. This way you don't damage the threads on the pinion gear coming out, and the pinion doesn't go flying out of the back of the carrier.